‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

‎For a set of non-negative integers~$L$‎, ‎the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...

For a set of non-negative integers L, the L-intersection number of a graph is the smallest number l for which there is an assignment of subsets Av ⊆ {1, . . . , l} to vertices v, such that every two vertices u, v are adjacent if and only if |Au ∩ Av| ∈ L. The bipartite L-intersection number is defined similarly when the conditions are considered only for the vertices in different parts. In this...

The intersection dimension of a bipartite graph with respect to a type L is the smallest number t for which it is possible to assign sets Ax ⊆ {1, . . . , t} of labels to vertices x so that any two vertices x and y from different parts are adjacent if and only if |Ax ∩Ay| ∈ L. The weight of such a representation is the sum ∑x |Ax| over all vertices x. We exhibit explicit bipartite n×n graphs wh...

the zarankiewicz number z(b; s) is the maximum size of a subgraph of kb,b which does not contain ks,s as a subgraph. the two-color bipartite ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of kb,b with two colors contains a ks,s in the rst color or a kt,t in the second color.in this work, we design and exploit a computational method for bounding and computin...

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideal...

1 Stable 2-Pairs and (X; Y)-Intersection Graphs 2 Abstract Given two xed graphs X and Y , the (X; Y)-intersection graph of a graph G is a graph where 1. each vertex corresponds to a distinct induced subgraph in G that is iso-morphic to Y , and 2. two vertices are adjacent ii the intersection of their corresponding sub-graphs contains an induced subgraph isomorphic to X. This notion generalizes ...

The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application o...

A universal representation theorem is derived that shows any graph is the intersection graph of one chordal graph, a number of co-bipartite graphs, and one unit interval graph. Central to the the result is the notion of the clique cover width which is a generalization of the bandwidth parameter. Specifically, we show that any planar graph is the intersection graph of one chordal graph, four co-...

We present new algebraic approaches for several well-known combinatorial problems, including non-bipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For non-bipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n) where n is the number of vertices and ω is the m...